Is the fast of a patient with atrophy of the nasal sinuses and migraines, who uses nasal drops and sometimes vomits, valid if she offers expiation for the entire month?

If nasal discharge does not reach the throat, then there is no problem with it and it does not invalidate the fast. However, if it reaches the throat, it breaks the fast according to the majority of scholars. The Islamic Fiqh Council, however, is of the opinion that it does not break the fast if the fasting person avoids swallowing what reaches the throat. The first opinion is more cautious.

As for vomiting, if it overcomes you, there is no issue with it. But if you intentionally induce vomiting, your fast is broken and you must make it up, due to the saying of the Prophet, peace and blessings be upon him: "Whoever is overcome by vomiting does not have to make it up, but whoever induces vomiting must make it up." You are not required to pay expiation, only to make up the fast.